MATH 3360-003
Exam II
Make Up
June 25, 2007
Answer the problems on separate paper. You do not need to rewrite the problem statements on
your answer sheets. Do your own work. Show all relevant steps which lead to your solutions.
For no question (except parts of Question 12) is True/Yes or False/No by itself sufficient as
the answer. Retain this question sheet for your records.
1. Not repeatable
2. (7 pts)
Let S be the plane \ and for points p , q ∈ S let d ( p , q ) denote the
(Euclidean) distance between p and q. Let O denote the origin, i.e., O = (0,0).
Let ~ be the relation on S given by p ~ q if d ( p , O ) = d ( q , O ) . Determine
whether ~ is an equivalence relation on S.
3A. (7 pts)
Let n be a positive integer and let a, b, c, d be integers. Suppose that
a ≡ b (mod n ) and c ≡ d (mod n ) . Prove that ac ≡ bd (mod n ) .
2
3B. (12 pts) For each of the following pairs of integers find the greatest common divisor and
the least common multiple:
a.
4. (7 pts)
{748,1360}
b.
{480,756}
For the following pair of integers find the greatest common divisor by using the
Euclidean algorithm:
a.
{332,544}
5. (7 pts)
Prove or disprove: ( a , b ) ≠ 1 and (b, c ) ≠ 1 ⇒ ( a , c ) ≠ 1
6. (6 pts)
Calculate the following value ([22] ⊕ [15]) : [11] in:
2
a.
]5
b.
] 13
c.
] 18
(Write the results in terms of the standard congruence classes [0], [1], . . . , [n-1]
for ] n .)
7. (7 pts)
Consider the group G = M ( Χ ) × ] 4 and the subgroup H = < ( μ180 , [1] ) > , where
the group M ( Χ ) is the set of invariant mappings (isometries) of the hour glass
and
μ180 is the 180D rotation about the center of the hour glass. Identify all of the
right cosets of H in G.
8. (6 pts)
Calculate each of the following:
a.
[ S5 :< (1 234) > ]
c.
⎡⎣ ] 6 × ] 12 :< [2] > ] 6 × < [2] > z12 ⎤⎦
[] 4 × ] 12 :< ([2],[2]) > ]
b.
9. (7 pts)
Construct the subgroup lattice of ] 24 .
10. (6 pts)
a.
In S 4 find < (1 4 3 2), (2 4) >
b.
In ] 24 find < [4], [10] >
11. (7 pts)
Prove: If A, B are abelian groups, then A × B is an abelian group.
12. (12 pts)
Determine whether the following pairs of groups are isomorphic or not. If the
pair of groups is isomorphic, you need only state so. If the pair of groups is not
isomorphic, you must give a valid reason why the pair of groups is not
isomorphic.
a.
] 3 × ] 6 , ] 18
b.
] 2 × ] 5 , ] 10
c.
< (3 5 7) > , G
d.
], E
e.
] 2 × ] 4 , M (, )
f.
]2 × ]4 × ]2, ]4 × ]2 × ]2
In part c. the group < (3 5 7) > is the subgroup of S 7 generated by the
permutation (3 5 7) and the group G is the group with operation given by the
Cayley table
*
a
b
c
a
c
a
b
b
a
b
c
c
b
c
a
In part d. the group E is the subgroup of ] of even integers.
In part e. the group M ( , ) is the set of invariant mappings (isometries) of the
square.
13. (6 pts)
Consider Abelian groups of order 48. Give a representative for each possible
isomorphism class.